Question

Find the quadratic polynomial whose zeroes are 1 and -3 . Verify the relationship between the coefficient and zeroes of polynomial

Answer 1

$\huge{\underline{\bf{\blue{Question:-}}}}$

Find the quadratic polynomial whose zeroes are 1 and -3 .

Verify the relationship between the coefficient and zeroes of polynomial.

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$\large{\underline{\bf{\pink{Answer:-}}}}$

Quadratic polynomial =x² + 2x -3 = 0

Relationship between coefficient and zeroes of polynomial is verified.

$\large{\underline{\bf{\purple{Explanation:-}}}}$

$\large{\underline{\bf{\green{Given:-}}}}$

Two zeroes are given as 1 and -3

$\large{\underline{\bf{\green{To\:Find:-}}}}$

we have to find the polynomial.

$\huge{\underline{\bf{\red{Solution:-}}}}$

Let α and β are zeroes of the polynomial.

let α = 1 and β =-3

➥ x² -(α+β)x +αβ = 0

➜ x² - [1+(-3)]x + 1× -3 = 0

➜ x² -(-2)x - 3 = 0

➜ x² + 2x -3 = 0

so the quadratic polynomial is x² + 2x -3 = 0.

Now verifying relation between zeroes of polynomial.

➥ α + β = - b/a

➜ 1 +(-3) = -2/1

➜ -2 = -2

So LHS = RHS .

Now,

➥ αβ = c/a

➜ 1×(-3) = -3/1

➜ -3 = -3

so, LHS = RHS

Hence relation is verified.

Answer 2

Explanation:

Given -

• zeroes are 1 and -3

To Find -

• A quadratic polynomial

Method 1 :-

Sum of zeroes :-

» 1 + (-3)

• » -2

And

Product of zeroes :-

» 1 × -3

• » -3

As we know that :-

For a quadratic polynomial :-

• x² - (sum of zeroes)x + (product of zeroes)

» x² - (-2)x + (-3)

» x² + 2x - 3

Hence,

The quadratic polynomial is x² + 2x - 3

Method 2 :-

As we know that :-

• α + β = -b/a

» 1 + (-3) = -b/a

» -2/1 = -b/a ....... (i)

And

• αβ = c/a

» 1 × -3 = c/a

» -3/1 = c/a ......... (ii)

From (i) and (ii), we get :

a = 1

b = 2

c = -3

And

As we know that :-

For a quadratic polynomial :-

• ax² + bx + c

It means,

» (1)x² + (2)x + (-3)

• » x² + 2x - 3

Hence,

The quadratic polynomial is x² + 2x - 3

Verification :-

• α + β = -b/a

» 1 + (-3) = -2/1

» -2 = -2

LHS = RHS

And

• αβ = c/a

» 1 × -3 = -3/1

» -3 = -3

LHS = RHS

Hence,

Verified..